Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

6. Uncertainty and Dynamics in the Collective model 261
which belongs to the interval [0, 1]. Note, moreover, that 0 < ¯ρ 0 (ys) < 1
unless one of the agents is (locally) risk neutral.
The first statement in Proposition 6.1 is often called the mutuality principle.
It states that when risk is shared efficiently, an agent’s consumption
is not affected by the idiosyncratic realization of her income; only shocks
affecting aggregate resources (here, total income ys) matter.Ithasbeen
used to test for efficient risk sharing, although the precise test is much
more complex than it may seem - we shall come back to this aspect below.
Formula (6.26) is quite interesting in itself. It can be rewritten as:
¯ρ 0 (ys) =
− u0a (¯ρ)
u 00a (¯ρ)
− u0a (¯ρ)
u 00a (¯ρ) − u0b (ys−¯ρ)
u 00b (ys−¯ρ)
(6.27)
The ratio − u0a (¯ρ)
u00a (¯ρ) is called the risk tolerance of A; it is the inverse of A’s
risk aversion. Condition (6.27) states that the marginal risk is allocated
between the agents in proportion of their respective risk tolerances. To
put it differently, assume the household’s total income fluctuates by one
(additional) dollar. The fraction of this one dollar fluctuation born by agent
a is proportional to a’s risk tolerance. To take an extreme case, if a was
infinitely risk averse - that is, her risk tolerance was nil - then ¯ρ 0 =0and
her share would remain constant: all the risk would be born by b.
It can actually be showed that the two conditions expressed by Proposition
6.1 are also sufficient. That is, take any sharing rule ρ satisfying them.
Then one can find two utility functions ua and ub such that ρ shares risk
efficiently between a and b. 9
6.3.3 Efficient risk sharing in a multi-commodity context: an
introduction
Regarding risk sharing, a multi commodity context is much more complex
than the one-dimensional world just described. The key insight is that consumption
decisions also depend on the relative prices of the various available
commodities, and that typically these prices fluctuate as well. Surprisingly
enough, sharing price risk is quite different from sharing income risk. A
precise investigation would be outside the scope of the present volume;
instead, we simply provide a short example. 10
9 The exact result is even slightly stronger; it states that for any ρ satisfying the
conditions and any increasing, strictly concave utility u A ,onecanfind some increasing,
strictly concave utility u B such that ρ shares risk efficiently between A and B (see Chiappori,
Samphantharak, Schulhofer-Wohl and Townsend 2010 for a precise statement).
10 The reader is referred to Chiappori, Townsend and Yamada (2008) for a precise
analysis. The following example is also borrowed from this article.